We solve the KleinGordon equation with the Morse empirical potential energy model. The bound state energy equation has been obtained in terms of the supersymmetric shape invariance approach. The relativistic vibrational transition frequencies for the X
^{1}
Σ
^{+}
state of ScI molecule have been computed by using the Morse potential model. The calculated relativistic vibrational transition frequencies are in good agreement with the experimental RKR values.
Introduction
There has been continuous interest in the analytical solutions of the KleinGordon equation with some typical diatomic molecule empirical potentials,
1
^{}
10
these potential models include the Morse potential,
11
RosenMorse potential,
12
ManningRosen potential,
13
PöschlTeller potential,
14
and DengFan potential,
15
etc. Diatomic potential energy functions have been applied in various issues, such as atomatom collisions, molecular spectroscopy, molecular dynamics simulation, chemical reactivity, and transport properties for more complex systems.
16
Alhaidari
et al
.
17
pointed out that one can yield a nonrelativistic limit with a potential function 2
V(r)
from the KleinGordon equation with the equal scalar potential
S(r)
and vector potential
V(r)
. In Ref.
18
and,
19
the authors investigated relativistic energy equations of the improved ManningRosen potential
20
and improved Rosen Morse potential,
21
and calculated relativistic vibrational transition frequencies for the a
^{3}
∑
_{u}
^{+}
state of
^{7}
Li
_{2}
molecule and the 3
^{3}
∑
_{g}
^{+}
state of Cs
_{2}
molecule.
In 1929, Morse
11
proposed the first threeparameter empirical potential energy function for diatomic molecules,
where
D_{e}
is the dissociation energy,
r_{e}
is the equilibrium bond length, and
α
denotes the range of the potential. The wellknown Morse potential has been widely used in many fields, such as the diatomic vibrations,
22
molecular simulations,
23
^{}
25
etc
. Based on the exact quantization rule for the nonrelativistic Schrödinger equation, Sun
1
proposed an exact quantization rule for the relativistic onedimensional KleinGordon equation, and obtained the exact relativistic energies for the onedimensional Morse potential. By using the asymptotic iteration method,
26
Bayrak
et al
.
2
investigated the bound state solutions of the KleinGordon equation for the equal scalar and vector Morse potentials. As far as we known, one has not reported predicting quantitatively the relativistic vibrational levels for real diatomic molecules in terms of the Morse potential model.
In this work, we attempt to study the bound state solutions of the KleinGordon equation with the Morse potential energy model. The bound state energy equation is investigated by using the basic concept of the supersymmetric shape invariance approach. We also calculate the relativistic vibrational transition frequencies of the X
^{1}
∑
^{+}
state of ScI molecule and compare the present calculated values with the RKR experimental data.
Bound State Solutions
The KleinGordon equation with a scalar potential
S
(
r
) and a vector potential
V
(
r
) is given by
where ∇
^{2}
is the Laplace operator,
E
is the relativistic energy of the quantum system,
M
is the rest mass of the quantum system, c is the speed of light, and
ħ=h/2π, h
is the Planck constant. The rest mass
M
can be taken as the reduced mass
μ
for a diatomic molecule. We write the wave function as ψ=(
r
,
θ
,
φ
)=(
u_{vJ}
(
r
)/
r
)
Y_{Jm}
(
θ
,
φ
), where
Y_{Jm}
(
θ
,
φ
) is the spherical harmonic function. Substituting this form of the wave function into Eq. (2), we reduce the radial part of the KleinGordon equation,
where
v
and
J
denotes the vibrational and rotational quantum numbers, respectively. In the presence of the equal scalar and vector potentials,
S
(
r
) =
V
(
r
), Eq. (3) turns to the following form
In the nonrelativistic limit, Eq. (4) produces the Schrödinger equation with the potential 2
V
(
r
). In the case of the nonrelativistic limit, we employ the scheme proposed by Alhaidari
et al
.
17
to make the interaction potential as
V
(
r
), not 2
V
(
r
). We write Eq. (4) in the form
Taking the scalar and vector potentials as the equal Morse potential,
S
(
r
) =
V
(
r
) =
U_{M}
(
r
), we produce the following secondorder Schrödingerlike equation,
This equation is exactly solvable for the case of
J
= 0. However, Eq. (6) is only approximately solvable when the centrifugal term is included (
J
≠0 ). We take the Pekeris approximation scheme to deal with the centrifugal term.
27
Taking the coordinate transformation of
x
= (
r
−
r_{e}
)/
r_{e}
, the centrifugal potential is expanded in a series around
x
= 0,
where
. The centrifugal potential
U_{CP}
(
r
) is replaced by the following form of
where
d
_{0}
,
d
_{1}
, and
d
_{2}
are the coefficients. Employing the coordinate transformation
x
= (
r
−
r_{e}
)/
r
_{e}
, we expand expression (8) in a series around
x
= 0,
Taking up to the secondorder degrees in the series expressions (7) and (9), and comparing equal powers of Eq. (7) and (9), we yield the following expressions for the coefficients
d
_{0}
,
d
_{1}
, and
d
_{2}
,
Substituting expression (8) into Eq. (6) gives the following equation
where
is defined as
We apply the supersymmetric shape invariance approach to solve Eq. (13).
28
^{}
30
The groundstate wave function
u
_{0,J}
(
r
) is expressed as
where
W
(
r
) is called a superpotential in supersymmetric quantum mechanics.
28
Substituting expression (15) into Eq. (13) leads us to obtain the following relation satisfied by the superpotential
W
(
r
),
where
is the groundstate energy. Letting the superpotential
W
(
r
) as
where
A
and
B
are two constants. Substituting this expression into expression (15) leads us to rewrite the groundstate wave function
u
_{0,J}
(
r
)as
We consider the bound state solutions, which demand the wave function
u_{vJ}(r)
to satisfy the boundary conditions:
u
_{vJ}
(∞)=0 and
u
_{vJ}
(0) is limitary. These regularity conditions demand
A
＞0 and
B
＞0.
Using expression (17) of superpotential
W
(
r
), we can construct a pair of supersymmetric partner potentials
U
_{−}
(
r
) and
U
_{+}
(
r
),
From expressions (19) and (20), we can have the following relationship
where
a
_{0}
=
B
,
a
_{1}
is a function of
a
_{0}
,
i.e
.,
a
_{1}
=
h
(
a
_{0}
) =
a
−
α
, and the reminder
R
(
a
_{1}
) is independent of
r
,
. Relation (21) shows that the partner potentials
U
_{−}
(
r
) and
U
_{+}
(
r
) are the shapeinvariant potentials. Their energy spectra can be determined with the shape invariance approach.
29
The energy spectra of the potential
U
_{−}
(
r
) are given by
where the quantum number ν = 0,1,2,….
By comparing Eq. (19) with Eq. (16), we obtain the following three relationships
Solving Eqs. (24) and (25), we have
From Eqs. (13), (16) and (19), we obtain the following relationship for
,
With the help of Eqs. (22) and (23), we arrive at the following expression for ,
Substituting expression (27) into expression (29) and using expression (26), we find the relativistic rotationvibrational energy equation for the diatomic molecule in the presence of equal scalar and vector Morse potential energy models,
When
J
= 0, 𝛾 = 0
s
, we obtain the relativistic vibrational energy equation for the diatomic molecule with equal scalar and vector Morse potentials,
Employing the superpotential
W
(
r
) given in expression (17) and the groundstate wave function
u
_{0,J}
(
r
) given in expression (18), we can calculate the excited state wave functions by using the explicit recursion operator approach.
31
32
Discussions
The force constant
k_{e}
is defined as the second derivates of the potential energy function
U
(
r
) for a diatomic molecule,
k_{e}
=(
d
^{2}
U
(
r
))/
dr
^{2}

_{r=re}
. From this definition and the relation
, we have the expression satisfied by the potential parameter
α
appearing in the Morse potential (1),
where
ω_{e}
is the equilibrium harmonic vibrational frequency.
We consider the X
^{1}
∑
^{+}
state of ScI molecule. Taking the experimental values of
D_{e}
,
r_{e}
, and
𝜔_{e}
as inputs, we can determine the value of the potential parameters
α
in terms of expression (32),
α
= 1.28094315 × 10
^{8}
cm
^{−1}
. The molecular constants of the X
^{1}
∑
^{+}
state of ScI molecule are taken from the literature
33
:
D_{e}
= 2.858 eV,
r_{e}
= 2.6078 Å, and
𝜔_{e}
= 277.18 cm
^{−1}
. A successful potential energy function should reproduce the experimental potential curve as determined by the RydbergKleinRees (RKR) method.
34
^{}
36
The experimental RKR data points reported by Reddy
et al
.
33
for the X
^{1}
∑
^{+}
state of ScI molecule are depicted in
Figure 1
, which also contains the potential energy curve reproduced by the Morse potential model. The average absolute deviation of the Morse potential for the X
^{1}
∑
^{+}
state of ScI molecule from the RKR potential reported by Reddy
et al
.
33
is 0.0344％ of
D_{e}
. This average absolute deviation satisfies the Lippincott criterion,
i.e
., an average absolute deviation of less than 1％ of the dissociation energy
D_{e}
.
37
The average deviation is defined as
where
N_{p}
is the number of experimental data points,
U
_{exp}
(
r
)and
U
_{calc}
(
r
) are the experimentally determined potential and the empirical potential, respectively. This accuracy indicator has been used in a large body of literature on assessing the accuracy of an empirical potential model.
38
39
(Color online) RKR data points and the Morse potential energy model for the X^{1}∑^{+} state of ScI molecule.
By applying energy eigenvalue Eq. (31), we can calculate relativistic vibrational transition frequencies for the X
^{1}
∑
^{+}
state of ScI molecule. The present calculated values are given in
Table 1
, in which we also list the RKR values taken from the literature.
33
It is clear that the relativistic vibrational transition frequencies obtained by using the Morse potential mode are good agreement with the RKR data.
A comparison of the calculated relativistic vibrational transition frequencies and experimental RKR values for the X1∑+state of ScI molecule (in units of cm−1)
A comparison of the calculated relativistic vibrational transition frequencies and experimental RKR values for the X^{1}∑^{+} state of ScI molecule (in units of cm^{−1})
Conclusions
In this work, we have studied the bound state solutions of the KleinGordon equation with the Morse potential energy model. The energy eigenvalue equation has been obtained in terms of the supersymmetric shape invariance approach. We calculate the relativistic vibrational transition frequencies for the X
^{1}
∑
^{+}
state of ScI molecule. The relativistic vibrational transition frequencies predicted with the Morse potential model are good agreement with the experimental RKR values.
Acknowledgements
This work was supported by the National Natural Science Foundation of China under Grant No. 10675097, and the Science Foundation of State Key Laboratory of Oil and Gas Reservoir Geology and Exploitation of China under Grant No. PLNZL001. And the publication cost of this paper was supported by the Korean Chemical Society.
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